Proper equilibrium in game theory is a refinement ofNash Equilibrium byRoger B. Myerson. Proper equilibrium further refinesReinhard Selten's notion of atrembling hand perfect equilibrium by assuming that more costly trembles are made with significantly smaller probability than lesscostly ones.

Given anormal form game and a parameter${\displaystyle ϵ>0}$, atotally mixed strategy profile${\displaystyle \sigma }$ is defined to be${\displaystyle ϵ}$-proper if, whenever a player has two pure strategies s and s' such that the expected payoff of playing s is smaller than the expected payoff of playing s' (that is), then the probability assigned to sis at most${\displaystyle ϵ}$ times the probability assigned to s'.

The strategy profile of the game is said to be a proper equilibriumif it is a limit point, as${\displaystyle ϵ}$ approaches 0, of a sequence of${\displaystyle ϵ}$-proper strategy profiles.

The game to the right is a variant ofMatching Pennies.

Player 1 (row player) hides a penny and if Player 2 (column player) guesses correctly whether it is heads up or tails up, he gets the penny. Inthis variant, Player 2 has a third option: grabbing the penny without guessing.TheNash equilibria of the game are the strategy profiles where Player 2 grabs the pennywith probability 1. Any mixed strategy of Player 1 is in (Nash) equilibrium with this pure strategyof Player 2. Any such pair is eventrembling hand perfect.Intuitively, since Player 1 expects Player 2 to grab the penny, he is not concerned aboutleaving Player 2 uncertain about whether it is heads up or tails up. However, it can be seenthat the unique proper equilibrium of this game is the one where Player 1 hides the penny heads up with probability 1/2 and tails up with probability 1/2 (and Player 2 grabs the penny). This unique proper equilibrium can be motivated intuitively as follows: Player 1 fully expects Player 2 to grab the penny.However, Player 1 still prepares for the unlikely event that Player 2 does not grab thepenny and instead for some reason decides to make a guess. Player 1 prepares for this event bymaking sure that Player 2 has no information about whether the penny is heads up or tails up,exactly as in the originalMatching Pennies game.

One may apply the properness notion toextensive form games in two different ways, completely analogous to the two different waystrembling hand perfectionis applied to extensive games. This leads to the notions ofnormal form proper equilibriumandextensive form proper equilibrium of an extensive form game. It was shown by vanDamme that a normal form proper equilibrium of an extensive form game is behaviorally equivalent toaquasi-perfect equilibrium of that game.

• Roger B. Myerson.Refinements of the Nash equilibrium concept.International Journal of Game Theory [de], 15:133-154, 1978.
• Eric van Damme. "A relationship between perfect equilibria in extensive form games and proper equilibria in normal form games."International Journal of Game Theory [de] 13:1--13, 1984.

• Game theory equilibrium concepts
• Non-cooperative games